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Mirrors > Home > MPE Home > Th. List > nn0ssz | Structured version Visualization version GIF version |
Description: Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nn0ssz | ⊢ ℕ0 ⊆ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-n0 11886 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnssz 11990 | . . 3 ⊢ ℕ ⊆ ℤ | |
3 | 0z 11980 | . . . 4 ⊢ 0 ∈ ℤ | |
4 | c0ex 10624 | . . . . 5 ⊢ 0 ∈ V | |
5 | 4 | snss 4679 | . . . 4 ⊢ (0 ∈ ℤ ↔ {0} ⊆ ℤ) |
6 | 3, 5 | mpbi 233 | . . 3 ⊢ {0} ⊆ ℤ |
7 | 2, 6 | unssi 4112 | . 2 ⊢ (ℕ ∪ {0}) ⊆ ℤ |
8 | 1, 7 | eqsstri 3949 | 1 ⊢ ℕ0 ⊆ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ∪ cun 3879 ⊆ wss 3881 {csn 4525 0cc0 10526 ℕcn 11625 ℕ0cn0 11885 ℤcz 11969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-i2m1 10594 ax-1ne0 10595 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 |
This theorem is referenced by: nn0z 11993 nn0zi 11995 nn0zd 12073 nn0ssq 12344 nthruz 15598 oddnn02np1 15689 evennn02n 15691 bitsf1ocnv 15783 pclem 16165 0ram 16346 0ram2 16347 0ramcl 16349 gexex 18966 iscmet3lem3 23894 plyeq0lem 24807 dgrlem 24826 2sqreultblem 26032 archirngz 30868 diophrw 39700 diophin 39713 diophun 39714 eq0rabdioph 39717 eqrabdioph 39718 rabdiophlem1 39742 diophren 39754 etransclem48 42924 |
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